Here's my fuel to the fire
I (think) I remember from an engineering class that a drill "bit" is actually just a drill, and the device used to drive the drill (if electric) is called a drill "motor." That means your hand held Dewalt, or lathe, or milling machine, or drill press, whatever, are all technically drill motors.
Although, as a machinist and engineer, even
I can't be that anal about terminology.
To the point: A person working in a hardware store should know what a twist drill is.